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Theorem xlt2add 9504
Description: Extended real version of lt2add 8074. Note that ltleadd 8075, which has weaker assumptions, is not true for the extended reals (since  0  + +oo  <  1  + +oo fails). (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
xlt2add  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( ( A  <  C  /\  B  <  D )  ->  ( A +e B )  <  ( C +e D ) ) )

Proof of Theorem xlt2add
StepHypRef Expression
1 xaddcl 9484 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  e.  RR* )
213ad2ant1 970 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  e.  RR* )
32adantr 272 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e B )  e.  RR* )
4 simp1l 973 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  e.  RR* )
5 simp2r 976 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  e.  RR* )
6 xaddcl 9484 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  D  e.  RR* )  ->  ( A +e D )  e.  RR* )
74, 5, 6syl2anc 406 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e D )  e.  RR* )
87adantr 272 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e D )  e.  RR* )
9 xaddcl 9484 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  D  e.  RR* )  ->  ( C +e D )  e.  RR* )
1093ad2ant2 971 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e D )  e.  RR* )
1110adantr 272 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( C +e D )  e.  RR* )
12 simp3r 978 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  <  D )
1312adantr 272 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  B  <  D )
14 simp1r 974 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  e.  RR* )
1514adantr 272 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR* )
165adantr 272 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR* )
17 simprl 501 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
18 xltadd2 9501 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  D  e.  RR*  /\  A  e.  RR )  ->  ( B  <  D  <->  ( A +e B )  <  ( A +e D ) ) )
1915, 16, 17, 18syl3anc 1184 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( B  <  D  <->  ( A +e B )  < 
( A +e
D ) ) )
2013, 19mpbid 146 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e B )  <  ( A +e D ) )
21 simp3l 977 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  <  C )
2221adantr 272 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  A  <  C )
234adantr 272 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR* )
24 simp2l 975 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  e.  RR* )
2524adantr 272 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR* )
26 simprr 502 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
27 xltadd1 9500 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  D  e.  RR )  ->  ( A  <  C  <->  ( A +e D )  <  ( C +e D ) ) )
2823, 25, 26, 27syl3anc 1184 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A  <  C  <->  ( A +e D )  < 
( C +e
D ) ) )
2922, 28mpbid 146 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e D )  <  ( C +e D ) )
303, 8, 11, 20, 29xrlttrd 9433 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e B )  <  ( C +e D ) )
3130anassrs 395 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  /\  D  e.  RR )  ->  ( A +e B )  < 
( C +e
D ) )
32 pnfxr 7690 . . . . . . . . . . . 12  |- +oo  e.  RR*
3332a1i 9 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> +oo  e.  RR* )
34 pnfge 9416 . . . . . . . . . . . 12  |-  ( C  e.  RR*  ->  C  <_ +oo )
3524, 34syl 14 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  <_ +oo )
364, 24, 33, 21, 35xrltletrd 9435 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  < +oo )
37 npnflt 9439 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A  < +oo  <->  A  =/= +oo )
)
384, 37syl 14 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A  < +oo  <->  A  =/= +oo )
)
3936, 38mpbid 146 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  =/= +oo )
40 pnfge 9416 . . . . . . . . . . . 12  |-  ( D  e.  RR*  ->  D  <_ +oo )
415, 40syl 14 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  <_ +oo )
4214, 5, 33, 12, 41xrltletrd 9435 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  < +oo )
43 npnflt 9439 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( B  < +oo  <->  B  =/= +oo )
)
4414, 43syl 14 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( B  < +oo  <->  B  =/= +oo )
)
4542, 44mpbid 146 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  =/= +oo )
46 xaddnepnf 9482 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )
)  ->  ( A +e B )  =/= +oo )
474, 39, 14, 45, 46syl22anc 1185 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  =/= +oo )
48 npnflt 9439 . . . . . . . . 9  |-  ( ( A +e B )  e.  RR*  ->  ( ( A +e
B )  < +oo  <->  ( A +e B )  =/= +oo ) )
492, 48syl 14 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( ( A +e B )  < +oo  <->  ( A +e B )  =/= +oo ) )
5047, 49mpbird 166 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  < +oo )
5150adantr 272 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( A +e
B )  < +oo )
52 oveq2 5714 . . . . . . 7  |-  ( D  = +oo  ->  ( C +e D )  =  ( C +e +oo ) )
53 mnfxr 7694 . . . . . . . . . . 11  |- -oo  e.  RR*
5453a1i 9 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  e.  RR* )
55 mnfle 9419 . . . . . . . . . . 11  |-  ( A  e.  RR*  -> -oo  <_  A )
564, 55syl 14 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <_  A
)
5754, 4, 24, 56, 21xrlelttrd 9434 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  C
)
58 nmnfgt 9442 . . . . . . . . . 10  |-  ( C  e.  RR*  ->  ( -oo  <  C  <->  C  =/= -oo )
)
5924, 58syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  C  <->  C  =/= -oo )
)
6057, 59mpbid 146 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  =/= -oo )
61 xaddpnf1 9470 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( C +e +oo )  = +oo )
6224, 60, 61syl2anc 406 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e +oo )  = +oo )
6352, 62sylan9eqr 2154 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( C +e
D )  = +oo )
6451, 63breqtrrd 3901 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( A +e
B )  <  ( C +e D ) )
6564adantlr 464 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  /\  D  = +oo )  ->  ( A +e B )  < 
( C +e
D ) )
66 simpr 109 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = -oo )  ->  D  = -oo )
67 mnfle 9419 . . . . . . . . . 10  |-  ( B  e.  RR*  -> -oo  <_  B )
6814, 67syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <_  B
)
6954, 14, 5, 68, 12xrlelttrd 9434 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  D
)
70 nmnfgt 9442 . . . . . . . . 9  |-  ( D  e.  RR*  ->  ( -oo  <  D  <->  D  =/= -oo )
)
715, 70syl 14 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  D  <->  D  =/= -oo )
)
7269, 71mpbid 146 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  =/= -oo )
7372adantr 272 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = -oo )  ->  D  =/= -oo )
7466, 73pm2.21ddne 2350 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = -oo )  ->  ( A +e
B )  <  ( C +e D ) )
7574adantlr 464 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  /\  D  = -oo )  ->  ( A +e B )  < 
( C +e
D ) )
76 elxr 9404 . . . . . 6  |-  ( D  e.  RR*  <->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
775, 76sylib 121 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
7877adantr 272 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  ->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
7931, 65, 75, 78mpjao3dan 1253 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  ->  ( A +e
B )  <  ( C +e D ) )
80 simpr 109 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = +oo )  ->  A  = +oo )
8139adantr 272 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = +oo )  ->  A  =/= +oo )
8280, 81pm2.21ddne 2350 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = +oo )  ->  ( A +e
B )  <  ( C +e D ) )
83 oveq1 5713 . . . . 5  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
84 xaddmnf2 9473 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
8514, 45, 84syl2anc 406 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo +e B )  = -oo )
8683, 85sylan9eqr 2154 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  ->  ( A +e
B )  = -oo )
87 xaddnemnf 9481 . . . . . . 7  |-  ( ( ( C  e.  RR*  /\  C  =/= -oo )  /\  ( D  e.  RR*  /\  D  =/= -oo )
)  ->  ( C +e D )  =/= -oo )
8824, 60, 5, 72, 87syl22anc 1185 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e D )  =/= -oo )
89 nmnfgt 9442 . . . . . . 7  |-  ( ( C +e D )  e.  RR*  ->  ( -oo  <  ( C +e D )  <-> 
( C +e
D )  =/= -oo ) )
9010, 89syl 14 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  ( C +e
D )  <->  ( C +e D )  =/= -oo ) )
9188, 90mpbird 166 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  ( C +e D ) )
9291adantr 272 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  -> -oo  <  ( C +e D ) )
9386, 92eqbrtrd 3895 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  ->  ( A +e
B )  <  ( C +e D ) )
94 elxr 9404 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
954, 94sylib 121 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
9679, 82, 93, 95mpjao3dan 1253 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  <  ( C +e D ) )
97963expia 1151 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( ( A  <  C  /\  B  <  D )  ->  ( A +e B )  <  ( C +e D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 929    /\ w3a 930    = wceq 1299    e. wcel 1448    =/= wne 2267   class class class wbr 3875  (class class class)co 5706   RRcr 7499   +oocpnf 7669   -oocmnf 7670   RR*cxr 7671    < clt 7672    <_ cle 7673   +ecxad 9398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-i2m1 7600  ax-0id 7603  ax-rnegex 7604  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-xadd 9401
This theorem is referenced by:  bldisj  12329
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